Rank-one transformations, odometers, and finite factors

• Published in: Israel journal of mathematics Vol. 255; no. 1; pp. 231 - 249
• Main Authors: Foreman, Matthew, Gao, Su, Hill, Aaron, Silva, Cesar E., Weiss, Benjamin
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01-06-2023; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Group Theory and Generalizations; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Permutations; Theoretical; Transformations
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-022-2451-y

In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d′) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.